Restoration of the Droplet Size Distribution Function when Solving the Kinetic Equation by the Moment Method

Abstract

Bulk condensation is a fairly common phenomenon in science and technology, so there is a need to determine the characteristics of a flow with condensing impurities, including the mass flow rate, quantity, and size of the formed and growing particles. To solve the problem of separating these particles from a gas–droplet flow, the type of distribution function, as well as the determination of the mass flow rate of droplets depending on their size, are the main initial data on which the quality of the design of the separation device depends. In the process of solving the kinetic equation using a direct numerical solution, the moments, as well as the particle size distribution function, are available at each step and do not require their reconstruction, whereas only the corresponding moments are available when using the moment method, but the distribution function is not. It is known that it is possible to reconstruct the form of the distribution function using the values of its moments, but the implementation of standard methods requires a large number of moments, and they have low stability or low accuracy. In this regard, the author’s modification of the well-known method for restoring the distribution function using the gamma distribution was developed, tested by comparison with known solutions. It is shown that the maximum difference between the known distribution function and the function reconstructed through a modification of the gamma distribution approximation, when solving the problem of bulk condensation during supersonic outflow of a vapor-gas mixture through a nozzle in a wide range of initial pressures, temperatures, and mass fractions of impurities for various components of the mixture, does not exceed 5%. In a subsequent comparison of the distribution function, reconstructed from the moments determined from the solution of the kinetic equation by the method of moments, with a direct numerical solution, the difference was at the level of 10%, while the probability of particles falling into a predetermined interval when ranking the distribution function by intervals was no more than 5%, which is acceptable for taking these data into account in flow separation problems. A method is proposed for determining the mass flow rate of droplets (particles) whose radius falls within a predetermined interval for the subsequent formulation of the problem of separating a gas–droplet or gas–dust flow using the reconstructed distribution function.